The Hilbert functions and the regularity of the graded components of local cohomology of a bigraded algebra are considered. Explicit bounds for these invariants are obtained for bigraded hypersurface rings. Introduction In this paper we study algebraic properties of the graded components of local cohomology of a bigradedK-algebra. Let P0 be a Noetherian ring, P = P0[y1, . . . , yn] be the polyn...

Using local cohomology and algebraic D-Modules, we generalize a comparison theorem between relative de Rham cohomology and Dwork cohomology due to N. Katz, A. Adolphson and S. Sperber.

Let R = ⊕n≥0Rn be a homogeneous noetherian ring and let M be a finitely generated graded R-module. Let H R+(M) denote the i-th local cohomology module of M with respect to the irrelevant ideal R+ := ⊕n>0Rn of R. We show that if R0 is a domain, there is some s ∈ R0\{0} such that the (R0)s-modules H R+(M)s are torsion-free (or vanishing) for all i. On use of this, we can deduce the following resu...

Let R = ⊕ n≥0 Rn be a Noetherian homogeneous ring with onedimensional local base ring (R0,m0). Let q0 ⊆ R0 be an m0-primary ideal, let M be a finitely generated graded R-module and let i ∈ N0. Let Hi R+ (M) denote the i-th local cohomology module of M with respect to the irrelevant ideal R+ := ⊕ n>0 Rn of R. We show that the first Hilbert-Samuel coefficient e1 ( q0, Hi R+ (M)n ) of the n-th gra...

We investigate the relative cohomology and relative homology theories of $F$-Gorenstein modules, consider the relations between classical and $F$-Gorenstein (co)homology theories.

Let S = K[x1, . . . , xn], let A, B be finitely generated graded S-modules, and let m = (x1, . . . , xn) ⊂ S. We give bounds for the regularity of the local cohomology of Tork (A, B) in terms of the graded Betti numbers of A and B, under the assumption that dim Tor1 (A, B) ≤ 1. We apply the results to syzygies, Gröbner bases, products and powers of ideals, and to the relationship of the Rees an...

We introduce a generalization of formal local cohomology module, which we call a formal local cohomology module with respect to a pair of ideals and study its various properties. We analyze their structure, the upper and lower vanishing and non-vanishing. There are various exact sequences concerning the formal cohomology modules. Among them a MayerVietoris sequence for two ideals with respect t...

In a remarkable paper Mats Boij and Jonas Söderberg [2006] conjectured that the Betti table of a Cohen-Macaulay module over a polynomial ring is a positive linear combination of Betti tables of modules with pure resolutions. We prove a strengthened form of their Conjectures. Applications include a proof of the Multiplicity Conjecture of Huneke and Srinivasan and a proof of the convexity of a fa...